Lecture 14: Spin glasses Outline: the EA and SK models heuristic theory dynamics I: using random...

Lecture 14: Spin glasses

Outline:• the EA and SK models• heuristic theory• dynamics I: using random matrices• dynamics II: using MSR

Random Ising model

So far we dealt with “uniform systems” Jij was the same for all pairs of neighbours.

Random Ising model

What if every Jij is picked (independently) from some distribution?

Random Ising model

We want to know the average of physical quantities (thermodynamicfunctions, correlation functions, etc) over the distribution of Jij’s.

Random Ising model

Today: a simple model with <Jij> = 0

Random Ising model

Today: a simple model with <Jij> = 0: spin glass

Simple model (Edwards-Anderson)

Jij[ ]av

= 0, Jij2

z(Jij = J ji)

Nearest-neighbour model with z neighbours

Jij[ ]av

= 0, Jij2

z(Jij = J ji)

note averages over different “samples” (1 sample = 1 realization of choices of Jij’s for all pairs (ij) indicated by [ … ]av

Jij[ ]av

= 0, Jij2

z(Jij = J ji)

E = − JijSi

∑ S j − hi

∑ Si

= − 12 JijSi

∑ S j − hi

∑ Si

Jij[ ]av

= 0, Jij2

z(Jij = J ji)

E = − JijSi

∑ S j − hi

∑ Si

= − 12 JijSi

∑ S j − hi

∑ Si

non-uniform J: anticipate nonuniform magnetization

mi = Si

Sherrington-Kirkpatrick model

Every spin is a neighbour of every other one: z = (N – 1)

N −1≈

Mean field theory is exact for this model

N −1≈

Mean field theory is exact for this model (but it is not simple)

Heuristic mean field theoryreplace total field on Si,

H i = JijS j

∑ (take hi = 0)

by its mean

H i = JijS j

∑ (take hi = 0)

by its mean

H i = JijS j

Jijm j

∑(take hi = 0)

by its meanand calculate mi as the average S of a single spin in field H:

H i = JijS j

Jijm j

∑(take hi = 0)

H i = JijS j

Jijm j

mi = tanh β Jijm j

∑ ⎛

⎝ ⎜ ⎜

⎠ ⎟ ⎟

(take hi = 0)

H i = JijS j

Jijm j

mi = tanh β Jijm j

∑ ⎛

⎝ ⎜ ⎜

⎠ ⎟ ⎟

no preference for mi > 0 or <0:

(take hi = 0)

H i = JijS j

Jijm j

mi = tanh β Jijm j

∑ ⎛

⎝ ⎜ ⎜

⎠ ⎟ ⎟

no preference for mi > 0 or <0: [mij]av = 0

(take hi = 0)

H i = JijS j

Jijm j

mi = tanh β Jijm j

∑ ⎛

⎝ ⎜ ⎜

⎠ ⎟ ⎟

if there are local spontaneous magnetizations mi ≠ 0,measure them by the order parameter (Edwards-Anderson)

(take hi = 0)

H i = JijS j

Jijm j

mi = tanh β Jijm j

∑ ⎛

⎝ ⎜ ⎜

⎠ ⎟ ⎟

if there are local spontaneous magnetizations mi ≠ 0,measure them by the order parameter (Edwards-Anderson)

(take hi = 0)

q = mi2

self-consistent calculation of q:

To compute q: Hi is a sum of many (seemingly) independent terms

To compute q: Hi is a sum of many (seemingly) independent terms=> Hi is Gaussian

To compute q: Hi is a sum of many (seemingly) independent terms=> Hi is Gaussian with variance

Jijm j

∑ ⎛

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎢ ⎢

⎥ ⎥av

= JijJik

∑ m jmk

⎣ ⎢ ⎢

⎦ ⎥ ⎥av

≈ [Jij2

∑ ]av[m j2]

Jijm j

∑ ⎛

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎢ ⎢

⎥ ⎥av

= JijJik

∑ m jmk

⎣ ⎢ ⎢

⎦ ⎥ ⎥av

≈ [Jij2

∑ ]av[m j2]

= [Jij2

∑ ]av q ≡ J 2q

Jijm j

∑ ⎛

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎢ ⎢

⎥ ⎥av

= JijJik

∑ m jmk

⎣ ⎢ ⎢

⎦ ⎥ ⎥av

≈ [Jij2

∑ ]av[m j2]

= [Jij2

∑ ]av q ≡ J 2q

2πJ 2q∫ tanh2 βH( )exp −

⎝ ⎜

⎠ ⎟

Jijm j

∑ ⎛

⎝ ⎜ ⎜

⎠ ⎟ ⎟

⎢ ⎢

⎥ ⎥av

= JijJik

∑ m jmk

⎣ ⎢ ⎢

⎦ ⎥ ⎥av

≈ [Jij2

∑ ]av[m j2]

= [Jij2

∑ ]av q ≡ J 2q

⎝ ⎜

⎠ ⎟ (solve for q)

spin glass transition:

⎝ ⎜

⎠ ⎟

⎝ ⎜

⎠ ⎟

expand in β:

⎝ ⎜

⎠ ⎟

expand in β:

2πJ 2q∫ βH − 1

3 (βH)3 +L[ ]2exp −

⎝ ⎜

⎠ ⎟

⎝ ⎜

⎠ ⎟

expand in β:

2πJ 2q∫ βH − 1

3 (βH)3 +L[ ]2exp −

⎝ ⎜

⎠ ⎟

2πJ 2q∫ (βH)2 − 2

3 (βH)4 +L[ ]exp −H 2

⎝ ⎜

⎠ ⎟

⎝ ⎜

⎠ ⎟

expand in β:

2πJ 2q∫ βH − 1

3 (βH)3 +L[ ]2exp −

⎝ ⎜

⎠ ⎟

2πJ 2q∫ (βH)2 − 2

3 (βH)4 +L[ ]exp −H 2

⎝ ⎜

⎠ ⎟

= β 2J 2q − 23 ⋅3β 4J 4q2 +L

⎝ ⎜

⎠ ⎟

expand in β:

2πJ 2q∫ βH − 1

3 (βH)3 +L[ ]2exp −

⎝ ⎜

⎠ ⎟

2πJ 2q∫ (βH)2 − 2

3 (βH)4 +L[ ]exp −H 2

⎝ ⎜

⎠ ⎟

= β 2J 2q − 23 ⋅3β 4J 4q2 +L

critical temperature: Tc = J

⎝ ⎜

⎠ ⎟

expand in β:

2πJ 2q∫ βH − 1

3 (βH)3 +L[ ]2exp −

⎝ ⎜

⎠ ⎟

2πJ 2q∫ (βH)2 − 2

3 (βH)4 +L[ ]exp −H 2

⎝ ⎜

⎠ ⎟

= β 2J 2q − 23 ⋅3β 4J 4q2 +L

below Tc:

β 2J 2 −1( )q ≈ 2q2 ⇒ q ≈ Tc − T

⎝ ⎜

⎠ ⎟

expand in β:

2πJ 2q∫ βH − 1

3 (βH)3 +L[ ]2exp −

⎝ ⎜

⎠ ⎟

2πJ 2q∫ (βH)2 − 2

3 (βH)4 +L[ ]exp −H 2

⎝ ⎜

⎠ ⎟

= β 2J 2q − 23 ⋅3β 4J 4q2 +L

below Tc:

β 2J 2 −1( )q ≈ 2q2 ⇒ q ≈ Tc − T

This heuristic theory is right up to this point, but wrong below Tc.

the trouble below Tc

In the ferromagnet, it was safe to approximate

H i = JijS j

∑ ≈ Jijm j

H i = JijS j

∑ ≈ Jijm j

because the next term in a systematic expansion in β,

H i = JijS j

∑ ≈ Jijm j

H i = JijS j

∑ = Jijm j

∑ − βmi Jij2

∑ (1− m j2) +L

H i = JijS j

∑ ≈ Jijm j

H i = JijS j

∑ = Jijm j

∑ − βmi Jij2

∑ (1− m j2) +L

was O(1/z).

H i = JijS j

∑ ≈ Jijm j

H i = JijS j

∑ = Jijm j

∑ − βmi Jij2

∑ (1− m j2) +L

was O(1/z). But here, the average of the 1st term is zero and you have to keep the second order term, the mean of which is of the order of the rms value of the first term.

H i = JijS j

∑ ≈ Jijm j

H i = JijS j

∑ = Jijm j

∑ − βmi Jij2

∑ (1− m j2) +L

Thouless-Anderson-Palmer (TAP) equations):

mi = tanh β Jijm j

∑ − β 2mi Jij2

∑ (1− m j2) + βhi

⎣ ⎢ ⎢

⎦ ⎥ ⎥

H i = JijS j

∑ ≈ Jijm j

H i = JijS j

∑ = Jijm j

∑ − βmi Jij2

∑ (1− m j2) +L

Thouless-Anderson-Palmer (TAP) equations):

mi = tanh β Jijm j

∑ − β 2mi Jij2

∑ (1− m j2) + βhi

⎣ ⎢ ⎢

⎦ ⎥ ⎥______________

Onsager correction to mean field

Dynamics (I: simple way)

Glauber dynamics:

dP({S},t)

2 1+ Si tanh βhi(t)( )[ ]i

∑ P(S1L − SiL SN )

− 12 1− Si tanh βhi(t)( )[ ]

∑ P(S1L SiL SN )

Glauber dynamics:

dP({S},t)

− 12 1− Si tanh βhi(t)( )[ ]

∑ P(S1L SiL SN )

d Si(t)

dt= − Si(t) + tanh β JijS j (t)

∑ ⎛

⎝ ⎜ ⎜

⎠ ⎟ ⎟

recall we derived from this

Glauber dynamics:

dP({S},t)

− 12 1− Si tanh βhi(t)( )[ ]

∑ P(S1L SiL SN )

d Si(t)

∑ ⎛

⎝ ⎜ ⎜

⎠ ⎟ ⎟

mean field:

Glauber dynamics:

dP({S},t)

− 12 1− Si tanh βhi(t)( )[ ]

∑ P(S1L SiL SN )

d Si(t)

∑ ⎛

⎝ ⎜ ⎜

⎠ ⎟ ⎟

mean field:

dt= −mi + tanh βH i[ ]

Dynamics I (continued)

= −mi + βH ilinearize (above Tc):

= −mi + βH i

≈ −mi + β Jij

∑ m j − β 2mi Jij2(1− q

∑ ) + βhi

linearize (above Tc):

use TAP:

= −mi + βH i

≈ −mi + β Jij

∑ m j − β 2mi Jij2(1− q

∑ ) + βhi

= −mi + β Jij

∑ m j − miβ2J 2 + βhi (q = 0, T > Tc )

use TAP:

= −mi + βH i

≈ −mi + β Jij

∑ m j − β 2mi Jij2(1− q

∑ ) + βhi

= −mi + β Jij

∑ m j − miβ2J 2 + βhi (q = 0, T > Tc )

use TAP:

In basis where J is diagonal:

= −mi + βH i

≈ −mi + β Jij

∑ m j − β 2mi Jij2(1− q

∑ ) + βhi

= −mi + β Jij

∑ m j − miβ2J 2 + βhi (q = 0, T > Tc )

use TAP:

dt= −mλ 1+ β 2J 2 − βJλ[ ] + βhλ

= −mi + βH i

≈ −mi + β Jij

∑ m j − β 2mi Jij2(1− q

∑ ) + βhi

= −mi + β Jij

∑ m j − miβ2J 2 + βhi (q = 0, T > Tc )

use TAP:

dt= −mλ 1+ β 2J 2 − βJλ[ ] + βhλ

susceptibility:

χλ =∂mλ (ω)

∂hλ (ω)=

1− iωτ 0 + β 2J 2 − βJλ

= −mi + βH i

≈ −mi + β Jij

∑ m j − β 2mi Jij2(1− q

∑ ) + βhi

= −mi + β Jij

∑ m j − miβ2J 2 + βhi (q = 0, T > Tc )

use TAP:

dt= −mλ 1+ β 2J 2 − βJλ[ ] + βhλ

instability (transition) reached when maximum eigenvalue

susceptibility:

χλ =∂mλ (ω)

∂hλ (ω)=

1− iωτ 0 + β 2J 2 − βJλ

= −mi + βH i

≈ −mi + β Jij

∑ m j − β 2mi Jij2(1− q

∑ ) + βhi

= −mi + β Jij

∑ m j − miβ2J 2 + βhi (q = 0, T > Tc )

use TAP:

dt= −mλ 1+ β 2J 2 − βJλ[ ] + βhλ

instability (transition) reached when maximum eigenvalue

(Jλ )max =1+ β 2J 2

susceptibility:

χλ =∂mλ (ω)

∂hλ (ω)=

1− iωτ 0 + β 2J 2 − βJλ

eigenvalue spectrum of a random matrix

For a dense random matrix with mean square element value J2/N,the eigenvalue density is “semicircular”:

ρ(Jλ ) =1

2πJ4J 2 − Jλ

ρ(Jλ ) =1

2πJ4J 2 − Jλ

(Jλ )max = 2 ⇒ β c =1so

ρ(Jλ ) =1

2πJ4J 2 − Jλ

(Jλ )max = 2 ⇒ β c =1so

local susceptibility

ρ(Jλ ) =1

2πJ4J 2 − Jλ

(Jλ )max = 2 ⇒ β c =1so

χ(ω) =1

Nχ ii

∑ (ω) =1

Nχ λ

∑ (ω) = dλρ∫ (λ )χ λ (ω)

ρ(Jλ ) =1

2πJ4J 2 − Jλ

(Jλ )max = 2 ⇒ β c =1so

χ(ω) =1

Nχ ii

∑ (ω) =1

Nχ λ

∑ (ω) = dλρ∫ (λ )χ λ (ω)

2πJdλ

4J 2 − λ2

1− iωτ 0 + β 2J 2 − βλ−2J

ρ(Jλ ) =1

2πJ4J 2 − Jλ

(Jλ )max = 2 ⇒ β c =1so

χ(ω) =1

Nχ ii

∑ (ω) =1

Nχ λ

∑ (ω) = dλρ∫ (λ )χ λ (ω)

4J 2 − x 2

y − x−2J

∫ = y − y 2 − 4J 2[ ]

2πJdλ

4J 2 − λ2

1− iωτ 0 + β 2J 2 − βλ−2J

ρ(Jλ ) =1

2πJ4J 2 − Jλ

(Jλ )max = 2 ⇒ β c =1so

χ(ω) =1

Nχ ii

∑ (ω) =1

Nχ λ

∑ (ω) = dλρ∫ (λ )χ λ (ω)

4J 2 − x 2

y − x−2J

∫ = y − y 2 − 4J 2[ ]

y =1− iωτ 0 + β 2J 2, x = βJλwith€

2πJdλ

4J 2 − λ2

1− iωτ 0 + β 2J 2 − βλ−2J

critical slowing down

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

(J = 1)

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

small ω:

(J = 1)

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

small ω:

χ−1(ω) ≈ T(1− iωτ )

(J = 1)

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

small ω:

χ−1(ω) ≈ T(1− iωτ ), τ ∝1

T − Tc

(J = 1)

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

small ω:

χ−1(ω) ≈ T(1− iωτ ), τ ∝1

T − Tc

(J = 1)

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

small ω:

χ−1(ω) ≈ T(1− iωτ ), τ ∝1

T − Tc

(J = 1)

but note: for the softest mode (with eigenvalue 2J)

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

small ω:

χ−1(ω) ≈ T(1− iωτ ), τ ∝1

T − Tc

(J = 1)

χλ =β

1− iωτ 0 + β 2J 2 − 2βJ

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

small ω:

χ−1(ω) ≈ T(1− iωτ ), τ ∝1

T − Tc

(J = 1)

χλ =β

1− iωτ 0 + β 2J 2 − 2βJ

(1− βJ)2 − iωτ 0

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

small ω:

χ−1(ω) ≈ T(1− iωτ ), τ ∝1

T − Tc

(J = 1)

χλ =β

1− iωτ 0 + β 2J 2 − 2βJ

(1− βJ)2 − iωτ 0

so its relaxation time diverges twice as strongly:

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

small ω:

χ−1(ω) ≈ T(1− iωτ ), τ ∝1

T − Tc

(J = 1)

χλ =β

1− iωτ 0 + β 2J 2 − 2βJ

(1− βJ)2 − iωτ 0

so its relaxation time diverges twice as strongly:

τ ∝ 1

(T − Tc )2

Dynamics II: using MSRUse a “soft-spin” SK model:

E[φ] = 12 r0φi

2 + 14 u0φi

∑ − 12 Jijφi

∑ φ j − hi

∑ φi

E[φ] = 12 r0φi

2 + 14 u0φi

∑ − 12 Jijφi

∑ φ j − hi

∑ φi

E[φ] = 12 r0φi

2 + 14 u0φi

∑ − 12 Jijφi

∑ φ j − hi

∑ φi

Langevin dynamics:

∂φi

∂t= −

∂ E[φ]( )∂φi

+ η i(t) = −r0φi − u0φi3 + Jijφ j

∑ + hi + η i(t)

E[φ] = 12 r0φi

2 + 14 u0φi

∑ − 12 Jijφi

∑ φ j − hi

∑ φi

Langevin dynamics:

∂φi

∂t= −

∂ E[φ]( )∂φi

∑ + hi + η i(t)

Generating functional:

E[φ] = 12 r0φi

2 + 14 u0φi

∑ − 12 Jijφi

∑ φ j − hi

∑ φi

Langevin dynamics:

∂φi

∂t= −

∂ E[φ]( )∂φi

∑ + hi + η i(t)

Z[J,h,θ] = DφD ˆ φ exp −S[φ, ˆ φ ,J,h,θ]( )∫

E[φ] = 12 r0φi

2 + 14 u0φi

∑ − 12 Jijφi

∑ φ j − hi

∑ φi

Langevin dynamics:

∂φi

∂t= −

∂ E[φ]( )∂φi

∑ + hi + η i(t)

Z[J,h,θ] = DφD ˆ φ exp −S[φ, ˆ φ ,J,h,θ]( )∫

S[φ, ˆ φ ,J,h,θ] = d∫ t T ˆ φ i2 + i ˆ φ i

∂φi

∂t+ r0φi + u0φi

3 − Jijφ j

∑ − hi

⎝ ⎜ ⎜

⎠ ⎟ ⎟− iθ iφi

⎣ ⎢ ⎢

⎦ ⎥ ⎥i

averaging over the Jij

exp i dt( ˆ φ iφ j + ˆ φ jφi)Jij∫( )ij

= exp −J 2

2Ndt d ′ t ˆ φ i(t) ˆ φ i(t')φ j (t)φ j (t') + ˆ φ i(t)φi( ′ t )φ j (t) ˆ φ j ( ′ t )( )∫

∑ ⎡

⎣ ⎢ ⎢

⎦ ⎥ ⎥

= exp −J 2

∑ ⎡

⎣ ⎢ ⎢

⎦ ⎥ ⎥

The exponent contains

C(t − ′ t ) =1

Nφi(t)φi(t '), R(t − ′ t )

∑ =i

Nφi(t) ˆ φ i(t'),

= exp −J 2

∑ ⎡

⎣ ⎢ ⎢

⎦ ⎥ ⎥

The exponent contains

C(t − ′ t ) =1

Nφi(t)φi(t '), R(t − ′ t )

∑ =i

Nφi(t) ˆ φ i(t'),

so replace them in the exponent

= exp − 12 J 2 dt d ′ t ˆ φ i(t) ˆ φ i(t')C(t − ′ t ) − i ˆ φ i(t)φi(t')R(t − ′ t )( )∫

∑ ⎡

⎣ ⎢

⎦ ⎥

decoupling sitesand introduce delta functions

1= DCD ˆ C ∫ exp − dtd ′ t ˆ C (t − ′ t ) NC(t − ′ t ) − φi(t)φi(t ')i

∑ ⎛

⎝ ⎜

⎠ ⎟∫

⎣ ⎢

⎦ ⎥

1= DRDCR∫ exp − dtd ′ t ˆ R (t − ′ t ) NR(t − ′ t ) − i φi(t) ˆ φ i(t ')i

∑ ⎛

⎝ ⎜

⎠ ⎟∫

⎣ ⎢

⎦ ⎥

decoupling sitesand introduce delta functions

1= DCD ˆ C ∫ exp − dtd ′ t ˆ C (t − ′ t ) NC(t − ′ t ) − φi(t)φi(t ')i

∑ ⎛

⎝ ⎜

⎠ ⎟∫

⎣ ⎢

⎦ ⎥

1= DRDCR∫ exp − dtd ′ t ˆ R (t − ′ t ) NR(t − ′ t ) − i φi(t) ˆ φ i(t ')i

∑ ⎛

⎝ ⎜

⎠ ⎟∫

⎣ ⎢

⎦ ⎥

We are left with

W ≡ Z[0,0,J][ ]av

= DC D ˆ C DR D ˆ R exp −N dt d ′ t ˆ C (t − ′ t )C(t − ′ t ) − ˆ R (t − ′ t )R(t − ′ t )[ ]∫( )∫

×exp N log DφD ˆ φ exp∫ −Sloc[φ, ˆ φ ,C, ˆ C ,R, ˆ R ]( )( )

(almost there)

Sloc[φ, ˆ φ ,C, ˆ C ,R, ˆ R ] = d∫ t T ˆ φ 2 + i ˆ φ ˙ φ + r0φ + u0φ3

( )[ ]

+ 12 J 2 dt d ′ t C(t − ′ t ) ˆ φ (t) ˆ φ ( ′ t ) − iR(t − ′ t ) ˆ φ (t)φ( ′ t )[ ]∫

+ dt d ′ t ˆ C (t − ′ t )φ(t)φ( ′ t ) − i ˆ R (t − ′ t )φ(t) ˆ φ ( ′ t )[ ]∫

(almost there)

( )[ ]

saddle-point equations:

(almost there)

( )[ ]

wrt ˆ C : C(t − t') = φ(t)φ( ′ t )loc

(almost there)

( )[ ]

wrt C : ˆ C (t − t') = − 12 J 2 ˆ φ (t) ˆ φ ( ′ t )

loc= 0

(almost there)

( )[ ]

loc= 0

wrt ˆ R : R(t − t') = i φ(t) ˆ φ ( ′ t )loc

(almost there)

( )[ ]

loc= 0

wrt ˆ R : R(t − t') = i φ(t) ˆ φ ( ′ t )loc

wrt R : ˆ R (t − ′ t ) = 12 iJ 2 ˆ φ (t)φ( ′ t )

loc= 1

2 J 2R( ′ t − t)

effective 1-spin problem:

The average correlation and response functions are equal to those of a self-consistent single-spin problem with action

( )[ ]

+ 12 J 2 dt d ′ t ˆ φ (t)C(t − ′ t ) ˆ φ ( ′ t )∫

−J 2 dt d ′ t ∫ i ˆ φ (t)R(t − ′ t )φ( ′ t )[ ]

( )[ ]

+ 12 J 2 dt d ′ t ˆ φ (t)C(t − ′ t ) ˆ φ ( ′ t )∫

−J 2 dt d ′ t ∫ i ˆ φ (t)R(t − ′ t )φ( ′ t )[ ]

describing a single spin

( )[ ]

+ 12 J 2 dt d ′ t ˆ φ (t)C(t − ′ t ) ˆ φ ( ′ t )∫

−J 2 dt d ′ t ∫ i ˆ φ (t)R(t − ′ t )φ( ′ t )[ ]

describing a single spin subject tonoise with correlation function 2Tδ(t – t’) +J2C(t - t’)

( )[ ]

+ 12 J 2 dt d ′ t ˆ φ (t)C(t − ′ t ) ˆ φ ( ′ t )∫

−J 2 dt d ′ t ∫ i ˆ φ (t)R(t − ′ t )φ( ′ t )[ ]

describing a single spin subject tonoise with correlation function 2Tδ(t – t’) +J2C(t - t’)and retarded self-interaction J2R(t - t’)

local response function

single effective spin obeys

dt= −r0S − u0S

3 + J 2 d ′ t −∞

∫ R(t − ′ t )S( ′ t ) + h(t) + ξ (t)

dt= −r0S − u0S

3 + J 2 d ′ t −∞

∫ R(t − ′ t )S( ′ t ) + h(t) + ξ (t)

ξ (t)ξ ( ′ t ) = 2Tδ(t − ′ t ) + J 2C(t − ′ t )

dt= −r0S − u0S

3 + J 2 d ′ t −∞

∫ R(t − ′ t )S( ′ t ) + h(t) + ξ (t)

ξ (t)ξ ( ′ t ) = 2Tδ(t − ′ t ) + J 2C(t − ′ t )

Fourier transform (u0 = 0)

dt= −r0S − u0S

3 + J 2 d ′ t −∞

∫ R(t − ′ t )S( ′ t ) + h(t) + ξ (t)

ξ (t)ξ ( ′ t ) = 2Tδ(t − ′ t ) + J 2C(t − ′ t )

−iωS(ω) = −r0S(ω) + J 2R0(ω)S(ω) + h(ω) + ξ (ω)

dt= −r0S − u0S

3 + J 2 d ′ t −∞

∫ R(t − ′ t )S( ′ t ) + h(t) + ξ (t)

ξ (t)ξ ( ′ t ) = 2Tδ(t − ′ t ) + J 2C(t − ′ t )

response function (susceptibility)

R0(ω) =∂ S(ω)

∂h(ω)=

r0 − J 2R0(ω)

dt= −r0S − u0S

3 + J 2 d ′ t −∞

∫ R(t − ′ t )S( ′ t ) + h(t) + ξ (t)

ξ (t)ξ ( ′ t ) = 2Tδ(t − ′ t ) + J 2C(t − ′ t )

response function (susceptibility)

R0(ω) =∂ S(ω)

∂h(ω)=

r0 − J 2R0(ω)

(Can solve quadratic equation for R0 to find it explicitly)

critical slowing downat small ω, R0

-1(ω) ~ 1 - iωτ

R0−1(ω) = r0 − J 2R0(ω)from

-1(ω) ~ 1 - iωτ

R0−1(ω) = r0 − J 2R0(ω)from

compute

τ =limω →0

∂R0−1(ω)

∂(−iω)

⎝ ⎜

⎠ ⎟=1+ J 2R0

2(0) limω →0

∂R0−1(ω)

∂(−iω)

⎝ ⎜

⎠ ⎟

⎣ ⎢

⎦ ⎥

-1(ω) ~ 1 - iωτ

R0−1(ω) = r0 − J 2R0(ω)from

compute

τ =limω →0

∂R0−1(ω)

∂(−iω)

⎝ ⎜

⎠ ⎟=1+ J 2R0

2(0) limω →0

∂R0−1(ω)

∂(−iω)

⎝ ⎜

⎠ ⎟

⎣ ⎢

⎦ ⎥

τ =1+ J 2R02(0)τ

-1(ω) ~ 1 - iωτ

R0−1(ω) = r0 − J 2R0(ω)from

compute

τ =limω →0

∂R0−1(ω)

∂(−iω)

⎝ ⎜

⎠ ⎟=1+ J 2R0

2(0) limω →0

∂R0−1(ω)

∂(−iω)

⎝ ⎜

⎠ ⎟

⎣ ⎢

⎦ ⎥

τ =1+ J 2R02(0)τ

⇒ τ =1

1− J 2R02(0)

R(0) =1

⎝ ⎜

⎠ ⎟

-1(ω) ~ 1 - iωτ

R0−1(ω) = r0 − J 2R0(ω)from

compute

τ =limω →0

∂R0−1(ω)

∂(−iω)

⎝ ⎜

⎠ ⎟=1+ J 2R0

2(0) limω →0

∂R0−1(ω)

∂(−iω)

⎝ ⎜

⎠ ⎟

⎣ ⎢

⎦ ⎥

τ =1+ J 2R02(0)τ

⇒ τ =1

1− J 2R02(0)

R(0) =1

⎝ ⎜

⎠ ⎟ critical slowing down

at Tc = J

-1(ω) ~ 1 - iωτ

R0−1(ω) = r0 − J 2R0(ω)from

compute

τ =limω →0

∂R0−1(ω)

∂(−iω)

⎝ ⎜

⎠ ⎟=1+ J 2R0

2(0) limω →0

∂R0−1(ω)

∂(−iω)

⎝ ⎜

⎠ ⎟

⎣ ⎢

⎦ ⎥

τ =1+ J 2R02(0)τ

⇒ τ =1

1− J 2R02(0)

R(0) =1

⎝ ⎜

⎠ ⎟ critical slowing down

at Tc = J

(u0 > 0: perturbation theory does not change this qualitatively)

Lecture 14: Spin glasses Outline: the EA and SK models heuristic theory dynamics I: using random...

Documents

MCMC Using Hamiltonian Dynamics

H3C MSR 900 Routers - Hewlett Packard Enterpriseh17007. · The H3C MSR 900 Routers Installation Guide describes how to install the H3C MSR 900 Routers, ... MSR Series Routers Command

MSR 2500 HR 1CT/2 - MSR Hot Restrike | PHILIPS

Msr Series

Cool-Tether: Energy Efficient On-the-fly WiFi Hot-spots using Mobile Smartphones 1 Vishnu Navda (MSR India) Ashish Sharma (MSR India Intern/UCSB) Ram Ramjee

1 Nick Harvey (MIT) Kamal Jain (MSR) Lap Chi Lau (U. Toronto) Chandra Nair (MSR) Yunnan Wu (MSR) Conservative Network Coding

MSR-net:Low-light Image Enhancement Using Deep ...MSR-net:Low-light Image Enhancement Using Deep Convolutional Network Liang Shen , Zihan Yue , Fan Feng, Quan Chen, Shihao Liu, and

Forecasting ﬁre dynamics using inverse Computational Fluid ... in... · Forecasting ﬁre dynamics using inverse Computational Fluid Dynamics modelling and Tangent Linearisation

MSR 175 Shock Transportation Data Logger: Protect your goods! · grams: MSR175 Dashboard, MSR ReportGenerator and MSR ShockViewer. The MSR Dashboard allows you to configure the loggers

Using Group Dynamics

MSR Format.doc

Msr presentation final

mop sinks model msr-2424 - FlorestoneModel MSR-2424 / MSR-3624 (specify) as manufactured by Florestone Products Co. Molding will be done under heat and pressure using matched metal

MSR Estimation Opinion

the MSR-INRIA Joint Centre 2006-2012pauillac.inria.fr/~levy/talks/12massoulie/massoulie.pdf · the MSR-INRIA Joint Centre msr-inria.inria.fr 2006-2012. Politics INRIA MSR Cambridge

MSH MSR 2014AppRiskManagementStrategiesMobEquip

tellabs 8800 msr

MSR 2011(PWD)

Raghu Meka (IAS, Princeton) Parikshit Gopalan (MSR, SVC) Omer Reingold (MSR, SVC)

MSR Engineering Documentation (MEDOC) Structure Principles ... · MSR Engineering Documentation (MEDOC) Structure Principles of MSRNET.DTD Scope: Automotive Networks MSR MEDOC, Dipl.-Ing-